I was wondering what are these two groups: $\mathbb{R}/\mathbb{Z}$ and $\mathbb{Q}/\mathbb{Z}$. Thanks!
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1Quotient groups – egreg Oct 03 '16 at 14:28
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1Are you asking about the notation, or what those groups are? That's two different questions. The notation is the standard "quotient group" notation. – Thomas Andrews Oct 03 '16 at 14:28
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What the groups are. – user336012 Oct 03 '16 at 14:33
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Possible links: here, here, here etc. – Dietrich Burde Oct 03 '16 at 15:46
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These are the quotient groups of $\mathbb{R}$ or $\mathbb{Q}$ by the subgroup $\mathbb{Z}$.
Starting with real numbers or rational numbers, declare two numbers equivalent if their difference is an integer. The equivalence classes under that relation form a group, called the quotient group. Using set-theoretic notation, we say $x \sim y$ if $x-y \in \mathbb{Z}$, and then $$ \mathbb{R}/\mathbb{Z} = \left\{[x] \mid x\in\mathbb{R}\right\} $$
More concretely, we cast off the “integer part” of every real/rational number. This turns out to be consistent with additions. For instance, if $[x]$ denotes the equivalent class of $x\in\mathbb{R}$ by $\mathbb{Z}$, then $$ [0.35] + [0.7] = [1.05] = [0.05] $$
Matthew Leingang
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What is the set notation for it, I think I would understand it better that way. For instance, R/Z = {.... | ...}. – user336012 Oct 03 '16 at 14:31
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Do you know about equivalence relations, @user336012 ? It takes a little effort to get a good set notation for this, unfortunately. – Thomas Andrews Oct 03 '16 at 14:37
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1Note that both these groups have a more common interpretation in terms of rotations of the circle. $\Bbb R/\Bbb Z$ is the full group of rotations, while $\Bbb Q/\Bbb Z$ is the group of rotations by a rational number of degrees. – Arthur Oct 03 '16 at 14:38
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